Idea

Given a groupGG and a subgroupHH, then their coset object is the quotientG/HG/H, hence the set of equivalence classes of elements of GG where two are regarded as equivalent if they differ by right multiplication with an element in HH.

Proof

Regarding the first statement:

Fix a unit vector in ℝn+1\mathbb{R}^{n+1}. Then its orbit under the defining O(n+1)O(n+1)-action on ℝn+1\mathbb{R}^{n+1} is clearly the canonical embedding Sn↪ℝn+1S^n \hookrightarrow \mathbb{R}^{n+1}. But precisely the subgroup of O(n+1)O(n+1) that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to O(n)O(n), hence Sn≃O(n+1)/O(n)S^n \simeq O(n+1)/O(n).

The second statement follows by the same kind of reasoning:

Clearly U(n+1)U(n+1)acts transitively on the unit sphereS2n+1S^{2n+1} in ℂn+1\mathbb{C}^{n+1}. It remains to see that its stabilizer subgroup of any point on this sphere is U(n)U(n). If we take the point with coordinates(1,0,0,⋯,0)(1,0, 0, \cdots,0) and regard elements of U(n+1)U(n+1) as matrices, then the stabilizer subgroup consists of matrices of the block diagonal form